Mathematics guidance: key stages 1 and 2 - Non-statutory guidance for the national curriculum in England Year 6 - Gov.uk
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Mathematics guidance: key stages 1 and 2 Non-statutory guidance for the national curriculum in England Year 6 June 2020
Summary
This document is one chapter of the full publication Mathematics guidance: key stages 1
and 2 Non-statutory guidance for the national curriculum in England.
An overview of the ready-to-progress criteria for all year groups is provided below,
followed by the specific guidance for year 6.
To find out more about how to use this document, please read the introductory chapter.
2Ready-to-progress criteria: year 1 to year 6
The table below is a summary of the ready-to-progress criteria for all year groups.
Strand Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
NPV 1NPV–1 Count within 3NPV–1 Know that 10 4NPV–1 Know that 10 5NPV–1 Know that 10 6NPV–1 Understand the
100, forwards and tens are equivalent to 1 hundreds are equivalent tenths are equivalent to relationship between
backwards, starting with hundred, and that 100 is to 1 thousand, and that 1 one, and that 1 is 10 powers of 10 from
any number. 10 times the size of 10; 1,000 is 10 times the size times the size of 0.1. 1 hundredth to 10 million,
apply this to identify and of 100; apply this to Know that 100 and use this to make a
work out how many 10s identify and work out how hundredths are equivalent given number 10, 100,
there are in other three- many 100s there are in to 1 one, and that 1 is 100 1,000, 1 tenth, 1
digit multiples of 10. other four-digit multiples times the size of 0.01. hundredth or 1
of 100. Know that 10 hundredths thousandth times the size
are equivalent to 1 tenth, (multiply and divide by 10,
and that 0.1 is 10 times 100 and 1,000).
the size of 0.01.
2NPV–1 Recognise the 3NPV–2 Recognise the 4NPV–2 Recognise the 5NPV–2 Recognise the 6NPV–2 Recognise the
place value of each digit place value of each digit place value of each digit place value of each digit place value of each digit
in two-digit numbers, and in three-digit numbers, in four-digit numbers, and in numbers with up to 2 in numbers up to 10
compose and decompose and compose and compose and decompose decimal places, and million, including decimal
two-digit numbers using decompose three-digit four-digit numbers using compose and decompose fractions, and compose
standard and non- numbers using standard standard and non- numbers with up to 2 and decompose numbers
standard partitioning. and non-standard standard partitioning. decimal places using up to 10 million using
partitioning. standard and non- standard and non-
standard partitioning. standard partitioning.
1NPV–2 Reason about 2NPV–2 Reason about 3NPV–3 Reason about 4NPV–3 Reason about 5NPV–3 Reason about 6NPV–3 Reason about
the location of numbers to the location of any two- the location of any three- the location of any four- the location of any the location of any
20 within the linear digit number in the linear digit number in the linear digit number in the linear number with up to 2 number up to 10 million,
number system, including number system, including number system, including number system, including decimals places in the including decimal
comparing using < > identifying the previous identifying the previous identifying the previous linear number system, fractions, in the linear
and = and next multiple of 10. and next multiple of 100 and next multiple of 1,000 including identifying the number system, and
and 10. and 100, and rounding to previous and next multiple round numbers, as
the nearest of each. of 1 and 0.1 and rounding appropriate, including in
to the nearest of each. contexts.
3Strand Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
NPV 3NPV–4 Divide 100 into 4NPV–4 Divide 1,000 into 5NPV–4 Divide 1 into 2, 6NPV–4 Divide powers of
2, 4, 5 and 10 equal parts, 2, 4, 5 and 10 equal 4, 5 and 10 equal parts, 10, from 1 hundredth to
and read scales/number parts, and read and read scales/number 10 million, into 2, 4, 5 and
lines marked in multiples scales/number lines lines marked in units of 1 10 equal parts, and read
of 100 with 2, 4, 5 and 10 marked in multiples of with 2, 4, 5 and 10 equal scales/number lines with
equal parts. 1,000 with 2, 4, 5 and 10 parts. labelled intervals divided
equal parts. into 2, 4, 5 and 10 equal
parts.
5NPV–5 Convert between
units of measure,
including using common
decimals and fractions.
NF 1NF–1 Develop fluency in 2NF–1 Secure fluency in 3NF–1 Secure fluency in
addition and subtraction addition and subtraction addition and subtraction
facts within 10. facts within 10, through facts that bridge 10,
continued practice. through continued
practice.
1NF–2 Count forwards 3NF–2 Recall 4NF–1 Recall 5NF–1 Secure fluency in
and backwards in multiplication facts, and multiplication and division multiplication table facts,
multiples of 2, 5 and 10, corresponding division facts up to and corresponding
up to 10 multiples, facts, in the 10, 5, 2, 4 , and recognise division facts, through
beginning with any and 8 multiplication products in multiplication continued practice.
multiple, and count tables, and recognise tables as multiples of the
forwards and backwards products in these corresponding number.
through the odd numbers. multiplication tables as
multiples of the
corresponding number.
4NF–2 Solve division
problems, with two-digit
dividends and one-digit
divisors, that involve
remainders, and interpret
remainders appropriately
according to the context.
3NF–3 Apply place-value 4NF–3 Apply place-value 5NF–2 Apply place-value
knowledge to known knowledge to known knowledge to known
additive and multiplicative additive and multiplicative additive and multiplicative
number facts (scaling number facts (scaling number facts (scaling
facts by 10). facts by 100) facts by 1 tenth or 1
hundredth).
4Strand Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
AS 1AS–1 Compose 2AS–1 Add and subtract 3AS–1 Calculate 6AS/MD–1 Understand
numbers to 10 from 2 across 10. complements to 100. that 2 numbers can be
parts, and partition related additively or
numbers to 10 into parts, multiplicatively, and
including recognising odd quantify additive and
and even numbers. multiplicative relationships
(multiplicative
relationships restricted to
multiplication by a whole
number).
1AS–2 Read, write and 2AS–2 Recognise the 3AS–2 Add and subtract 6AS/MD–2 Use a given
interpret equations subtraction structure of up to three-digit numbers additive or multiplicative
containing addition ( ), ‘difference’ and answer using columnar methods. calculation to derive or
subtraction ( ) and questions of the form, complete a related
equals ( ) symbols, and “How many more…?”. calculation, using
relate additive arithmetic properties,
expressions and inverse relationships, and
equations to real-life place-value
contexts. understanding.
2AS–3 Add and subtract 3AS–3 Manipulate the 6AS/MD–3 Solve
within 100 by applying additive relationship: problems involving ratio
related one-digit addition Understand the inverse relationships.
and subtraction facts: add relationship between
and subtract only ones or addition and subtraction,
only tens to/from a two- and how both relate to the
digit number. part–part–whole structure.
Understand and use the
commutative property of
addition, and understand
the related property for
subtraction.
2AS–4 Add and subtract 6AS/MD–4 Solve
within 100 by applying problems with 2
related one-digit addition unknowns.
and subtraction facts: add
and subtract any 2 two-
digit numbers.
5Strand Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
MD 2MD–1 Recognise 3MD–1 Apply known 4MD–1 Multiply and 5MD–1 Multiply and For year 6, MD ready-to-
repeated addition multiplication and division divide whole numbers by divide numbers by 10 and progress criteria are
contexts, representing facts to solve contextual 10 and 100 (keeping to 100; understand this as combined with AS ready-
them with multiplication problems with different whole number quotients); equivalent to making a to-progress criteria
equations and calculating structures, including understand this as number 10 or 100 times (please see above).
the product, within the 2, quotitive and partitive equivalent to making a the size, or 1 tenth or 1
5 and 10 multiplication division. number 10 or 100 times hundredth times the size.
tables. the size.
2MD–2 Relate grouping 4MD–2 Manipulate 5MD–2 Find factors and
problems where the multiplication and division multiples of positive whole
number of groups is equations, and numbers, including
unknown to multiplication understand and apply the common factors and
equations with a missing commutative property of common multiples, and
factor, and to division multiplication. express a given number
equations (quotitive as a product of 2 or 3
division). factors.
4MD–3 Understand and 5MD–3 Multiply any
apply the distributive whole number with up to
property of multiplication. 4 digits by any one-digit
number using a formal
written method.
5MD–4 Divide a number
with up to 4 digits by a
one-digit number using a
formal written method,
and interpret remainders
appropriately for the
context.
6Strand Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
F 3F–1 Interpret and write 6F–1 Recognise when
proper fractions to fractions can be
represent 1 or several simplified, and use
parts of a whole that is common factors to
divided into equal parts. simplify fractions.
3F–2 Find unit fractions of 5F–1 Find non-unit 6F–2 Express fractions in
quantities using known fractions of quantities. a common denomination
division facts and use this to compare
(multiplication tables fractions that are similar
fluency). in value.
3F–3 Reason about the 4F–1 Reason about the 6F–3 Compare fractions
location of any fraction location of mixed with different
within 1 in the linear numbers in the linear denominators, including
number system. number system. fractions greater than 1,
using reasoning, and
choose between
reasoning and common
denomination as a
comparison strategy.
4F–2 Convert mixed 5F–2 Find equivalent
numbers to improper fractions and understand
fractions and vice versa. that they have the same
value and the same
position in the linear
number system.
3F–4 Add and subtract 4F–3 Add and subtract 5F–3 Recall decimal
fractions with the same improper and mixed fraction equivalents for
denominator, within 1. fractions with the same
, , and , and
denominator, including
bridging whole numbers. for multiples of these
proper fractions.
G 1G–1 Recognise common 2G–1 Use precise 3G–1 Recognise right 5G–1 Compare angles,
2D and 3D shapes language to describe the angles as a property of estimate and measure
presented in different properties of 2D and 3D shape or a description of angles in degrees (°) and
orientations, and know shapes, and compare a turn, and identify right draw angles of a given
that rectangles, triangles, shapes by reasoning angles in 2D shapes size.
cuboids and pyramids are about similarities and presented in different
not always similar to one differences in properties. orientations.
another.
7Strand Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
G 5G–2 Compare areas and
calculate the area of
rectangles (including
squares) using standard
units.
1G–2 Compose 2D and 3G–2 Draw polygons by 4G–1 Draw polygons, 6G–1 Draw, compose,
3D shapes from smaller joining marked points, specified by coordinates and decompose shapes
shapes to match an and identify parallel and in the first quadrant, and according to given
example, including perpendicular sides. translate within the first properties, including
manipulating shapes to quadrant. dimensions, angles and
place them in particular area, and solve related
orientations. problems.
4G–2 Identify regular
polygons, including
equilateral triangles and
squares, as those in
which the side-lengths
are equal and the angles
are equal. Find the
perimeter of regular and
irregular polygons.
4G–3 Identify line
symmetry in 2D shapes
presented in different
orientations. Reflect
shapes in a line of
symmetry and complete a
symmetric figure or
pattern with respect to a
specified line of
symmetry.
8Year 6 guidance
Ready-to-progress criteria
Year 5 conceptual Year 6 ready-to- Key stage 3 applications
prerequesite progress criteria
Understand the relationship 6NPV–1 Understand the Understand and use place
between powers of 10 from 1 relationship between value for decimals,
hundredth to 1,000 in terms of powers of 10 from measures, and integers of
grouping and exchange (for 1 hundredth to 10 any size.
example, 1 is equal to 10 million, and use this to Interpret and compare
tenths) and in terms of scaling make a given number numbers in standard form
(for example, 1 is ten times the 10, 100, 1,000, 1 tenth, , where n is
size of 1 tenth). 1 hundredth or 1 a positive or negative
thousandth times the integer or zero.
size (multiply and divide
by 10, 100 and 1,000).
Recognise the place value of 6NPV–2 Recognise the Understand and use place
each digit in numbers with place value of each digit value for decimals,
units from thousands to in numbers up to 10 measures, and integers of
hundredths and compose and million, including any size.
decompose these numbers decimal fractions, and Order positive and negative
using standard and non- compose and integers, decimals, and
standard partitioning. decompose numbers up fractions.
to 10 million using
standard and non- Use a calculator and other
standard partitioning. technologies to calculate
results accurately and then
interpret them appropriately.
Reason about the location of 6NPV–3 Reason about Order positive and negative
numbers between 0.01 and the location of any integers, decimals, and
9,999 in the linear number number up to 10 million, fractions; use the number
system. including decimal line as a model for ordering
Round whole numbers to the fractions, in the linear of the real numbers; use the
nearest multiple of 1,000, 100 number system, and symbols =, ≠, , ≤, ≥
or 10, as appropriate. round numbers, as Round numbers and
appropriate, including in measures to an appropriate
Round decimal fractions to the contexts.
nearest whole number or degree of accuracy (for
nearest multiple of 0.01 example, to a number of
decimal places or significant
figures).
Use approximation through
rounding to estimate
answers and calculate
possible resulting errors
expressed using inequality
notation a < x ≤ b
9Year 5 conceptual Year 6 ready-to- Key stage 3 applications
prerequesite progress criteria
Divide 1000, 100 and 1 into 2, 6NPV–4 Divide powers Use standard units of mass,
4, 5 and 10 equal parts, and of 10, from 1 hundredth length, time, money, and
read scales/number lines with to 10 million, into 2, 4, 5 other measures, including
2, 4, 5 and 10 equal parts. and 10 equal parts, and with decimal quantities.
read scales/number Construct and interpret
lines with labelled appropriate tables, charts,
intervals divided into 2, and diagrams.
4, 5 and 10 equal parts.
Be fluent in all key stage 2 6AS/MD–1 Understand Understand that a
additive and multiplicative that 2 numbers can be multiplicative relationship
number facts (see Appendix: related additively or between 2 quantities can be
number facts fluency multiplicatively, and expressed as a ratio or a
overview quantify additive and fraction.
) and calculation. multiplicative Express 1 quantity as a
relationships fraction of another, where
Manipulate additive equations, (multiplicative
including applying the fraction is less than 1
relationships restricted and greater than 1.
understanding of the inverse to multiplication by a
relationship between addition whole number). Interpret mathematical
and subtraction, and the relationships both
commutative property of algebraically and
addition. geometrically.
Manipulate multiplicative Interpret when the structure
equations, including applying of a numerical problem
understanding of the inverse requires additive,
relationship between multiplicative or proportional
multiplication and division, and reasoning.
the commutative property of
multiplication.
Make a given number (up to 6AS/MD–1 Use a given Recognise and use
9,999, including decimal additive or multiplicative relationships between
fractions) 10, 100, 1 tenth or 1 calculation to derive or operations including inverse
hundredth times the size complete a related operations.
(multiply and divide by 10 and calculation, using Use algebra to generalise
100). arithmetic properties, the structure of arithmetic,
Apply place-value knowledge inverse relationships, including to formulate
to known additive and and place-value mathematical relationships.
multiplicative number facts understanding.
Understand and use
(scaling facts by 10, 100, 1 standard mathematical
tenth or 1 hundredth). formulae; rearrange
Manipulate additive equations. formulae to change the
Manipulate multiplicative subject.
equations.
10Year 5 conceptual Year 6 ready-to- Key stage 3 applications
prerequesite progress criteria
Recall multiplication and 6AS/MD–3 Solve Use ratio notation, including
division facts up to problems involving ratio reduction to simplest form.
. relationships. Divide a given quantity into
Apply place-value knowledge 2 parts in a given part:part
to known additive and or part:whole ratio; express
multiplicative number facts. the division of a quantity into
2 parts as a ratio.
Be fluent in all key stage 2 6AS/MD–4 Solve Reduce a given linear
additive and multiplicative problems with 2 equation in two variables to
number facts and calculation. unknowns. the standard form
Manipulate additive equations. y = mx + c; calculate and
interpret gradients and
Manipulate multiplicative intercepts of graphs of such
equations. linear equations
Find a fraction of a quantity. numerically, graphically and
algebraically.
Use linear and quadratic
graphs to estimate values of
y for given values of x and
vice versa and to find
approximate solutions of
simultaneous linear
equations.
Recall multiplication and 6F–1 Recognise when Use the concepts and
division facts up to fractions can be vocabulary of prime
. simplified, and use numbers, factors (or
common factors to divisors), multiples, common
Find factors and multiples of simplify fractions. factors, common multiples,
positive whole numbers, highest common factor,
including common factors and lowest common multiple,
common multiples. prime factorisation,
Find equivalent fractions and including using product
understand that they have the notation and the unique
same value and the same factorisation property.
position in the linear number Simplify and manipulate
system. algebraic expressions by
taking out common factors.
11Year 5 conceptual Year 6 ready-to- Key stage 3 applications
prerequesite progress criteria
Recall multiplication and 6F–2 Express fractions Order positive and negative
division facts up to in a common integers, decimals and
. denomination and use fractions.
this to compare Use the 4 operations,
Find factors and multiples of fractions that are similar
positive whole numbers. including formal written
in value. methods, applied to
Find equivalent fractions. integers, decimals, proper
Reason about the location of and improper fractions, and
fractions and mixed numbers mixed numbers, all both
in the linear number system. positive and negative.
Use and interpret algebraic
notation, including:
a/b in place of
coefficients written as
fractions rather than as
decimals.
Reason about the location of 6F–3 Compare fractions Order positive and negative
fractions and mixed numbers with different integers, decimals, and
in the linear number system. denominators, including fractions; use the number
Find equivalent fractions. fractions greater than 1, line as a model for ordering
using reasoning, and of the real numbers; use the
choose between symbols =, ≠, , ≤, ≥
reasoning and common
denomination as a
comparison strategy.
Find the perimeter of regular 6G–1 Draw, compose, Draw shapes and solve
and irregular polygons. and decompose shapes more complex geometry
Compare angles, estimate and according to given problems (see Mathematics
measure angles in degrees (°) properties, including programmes of study: key
and draw angles of a given dimensions, angles and stage 3 - Geometry and
size. area, and solve related measures).
problems.
Compare areas and calculate
the area of rectangles
(including squares) using
standard units.
126NPV–1 Powers of 10
Understand the relationship between powers of 10 from 1 hundredth to 10 million, and
use this to make a given number 10, 100, 1,000, 1 tenth, 1 hundredth or 1 thousandth
times the size (multiply and divide by 10, 100 and 1,000).
6NPV–1 Teaching guidance
An understanding of the relationship between the powers of 10 prepares pupils for
working with much larger or smaller numbers at key stage 3, when they will learn to read
and write numbers in standard form (for example, 600,000,000= 6 × 108 ).
Pupils need to know that what they learnt in year 3 and year 4 about the relationship
between 10, 100 and 1,000 (see 3NPV–1 and 4NPV–1), and in year 5 about the
relationship between 1, 0.1 and 0.01 (5NPV–1) extends through the number system. By
the end of year 6, pupils should have a cohesive understanding of the whole place-value
system, from decimal fractions through to 7-digit numbers.
Pupils need to be able to read and write numbers from 1 hundredth to 10 million, written
in digits, beginning with the powers or 10, as shown below, and should understand the
relationships between these powers of 10.
0. 01 one hundredth
0. 1 one tenth
1 one
1 0 ten
1 0 0 one hundred
1 , 0 0 0 one thousand
1 0 , 0 0 0 ten thousand
1 0 0 , 0 0 0 one hundred thousand
1 , 0 0 0 , 0 0 0 one million
10, 0 0 0 , 0 0 0 ten million
Pupils should know that each power of 10 is equal to 1 group of 10 of the next smallest
power of 10, for example 1 million is equal to 10 hundred thousands.
Figure 1: ten 100,000-value place-value counters in a tens frame
13Language focus
“10 hundred-thousands is equal to 1 million.”
Pupils should also understand this relationship in terms of scaling by 10 or one-tenth.
Language focus
“1,000,000 is 10 times the size of 100,000.”
“100,000 is one-tenth times the size of 1,000,000.”
Pupils must also understand the relationships between non-adjacent powers of 10 up to
a scaling by 1,000 or 1 thousandth (or grouping of up to 1,000 of a given power).
Language focus
“10 thousands is equal to 10,000.”
“10,000 is 10 times the size of 1,000.”
“1,000 is one-tenth times the size of 10,000.”
Pupils must also be able to write multiples of these powers of 10, including when there
are more than 10 of given power of 10, for example, 18 hundred thousands is written as
1,800,000. Pupils should be able to restate the quantity in the appropriate power of 10,
for example 18 hundred thousands is equal to 1 million 8 hundred thousand.
Once pupils understand the relationships between powers of ten, they should extend this
to other numbers in the Gattegno chart. They must be able to identify the number that is
10, 100, 1,000, 1 tenth, 1 hundredth or 1 thousandth times the size of a given number,
and associate this with multiplying or dividing by 10, 100 and 1,000. This will prepare
pupils for multiplying by decimals in key stage 3, when they will learn, for example, that
dividing by 100 is equivalent to multiplying by 0.01.
14Figure 2: using the Gattegno chart to multiply and divide by 100
Language focus
“50,000 is 100 times the size of 500.”
“500 multiplied by 100 is equal to 50,000.”
“500 is one-hundredth times the size of 50,000.”
“50,000 divided by 100 is equal to 500.”
Pupils should recognise the inverse relationship between, for example making a number
100 times the size, and returning to the original number by making it one-hundredth times
the size.
This understanding should then be extended to multiplicative calculations that involve
numbers with more than one significant digit, extending what pupils learnt in 5MD–1
about multiplying and dividing by 10 and 100.
1,659 × 100 165,900 165,900 ÷ 100 1,659
=
21,156 × 10 211,560 =
211,560 ÷ 10 21,156
47.1× 1,000 47,100
= 47,100 ÷ 1,000 47.1
=
Pupils can use the Gattegno chart for support throughout this criterion, but by the end of
year 6 they must be able to calculate without it.
You can find out more about fluency and recording for these calculations here in the
calculation and fluency section: Number, place value and number facts: 6NPV–1 and
6NPV–2
15Making connections
Writing multiples of powers of 10 depends on 6NPV–2. In 6AS/MD–2 pupils use their
understanding of place-value and scaling number facts to manipulate equations.
6NPV–1 Example assessment questions
Complete the sentences.
a. 500 made 1,000 times the size is .
b. 0.7 made 100 times the size is .
c. 800,000 made 10 times the size is .
d. 4,000,000 made one-thousandth times the size is .
e. 9,000 made one-hundredth times the size is .
f. 3 made one-tenth times the size is .
The distance from London to Bristol is about 170km. The distance from London to
Sydney, Australia is about 100 times as far. Approximately how far is it from London
to Sydney?
A newborn elephant weighs about 150kg. A newborn kitten weighs about 150g. How
many times the mass of a newborn kitten is a newborn elephant?
Walid has a place-value chart and three counters. He has represented the number
1,110,000.
a. Find 2 different numbers that Walid could make so that 1 number is one-hundredth
times the size of the other number.
b. Find 2 different numbers that Walid could make so that 1 number is 1,000 times
the size of the other number.
Fill in the missing numbers.
×10 ×10
→ →
4.3 27,158
← ←
÷10 ÷10
16×100 ×100
→ →
729 5,806
← ←
÷100 ÷100
×1,000 ×1,000
→ →
14.3 2,670,000
← ←
÷1,000 ÷1,000
Use the following to complete the equations:
×10 ×100 ×1,000 ÷10 ÷100 ÷1,000
Use each term only once.
543 = 5.43 3,169 = 3,169,000 515 = 5,150
276,104 = 27,610.4 35,000 = 35 427 = 42,700
6NPV–2 Place value in numbers up to 10,000,000
Recognise the place value of each digit in numbers up to 10 million, including decimal
fractions, and compose and decompose numbers up to 10 million using standard and
non-standard partitioning.
6NPV–2 Teaching guidance
Pupils must be able to read and write numbers up to 10,000,000, including decimal
fractions. Pupils should be able to use a separator (such as a comma) every third digit
from the decimal separator to help read and write numbers. Pupils must be able to copy
numbers from calculator displays, inserting thousands separators and decimal points
correctly. This will prepare them for secondary school, where pupils will be expected to
know how to use calculators.
Pupils need to be able to identify the place value of each digit in a number.
17Language focus
“In 67,000.4…
• the 6 represents 6 ten-thousands; the value of the 6 is 60,000
• the 7 represents 7 thousands; the value of the 7 is 7,000
• the 4 represents 4 tenths; the value of the 4 is 0.4”
Pupils must be able to combine units from millions to hundredths to compose numbers,
and partition numbers into these units, and solve related addition and subtraction
calculations. Pupils need to experience variation in the order of presentation of the units,
so that they understand, for example, that 5,034,000.2 is equal to 4,000 + 30,000 + 0.2 +
5,000,000. Pupils should be able to represent a given number in different ways, including
using place-value counters and Gattegno charts, and write numbers shown using these
representations.
Pupils should then have sufficient understanding of the composition of large numbers to
compare and order them by size.
Pupils also need to be able to solve problems relating to subtraction of any single place-
value part from a number, for example:
381,920 – 900 =
381,920 – = 380,920
As well as being able to partition numbers in the ‘standard’ way (into individual place-
value units), pupils must also be able to partition numbers in ‘non-standard’ ways, and
carry out related addition and subtraction calculations, for example:
518.32 + 30 =
548.32
381,920 – 60,000 = 321,920
Pupils can initially use place-value counters for support with this type of partitioning and
calculation, but by the end of year 6 must be able to partition and calculate without them.
You can find out more about fluency and recording for these calculations here in the
calculation and fluency section: Number, place value and number facts: 6NPV–1 and
6NPV–2
186NPV–2 Example assessment questions
What is the value of the digit 5 in each of these numbers?
a. 720,541
b. 5,876,023
c. 1,587,900
d. 651,920
e. 905,389
f. 2,120,806.50
g. 8,002,345
h. 701,003.15
Write a seven-digit number that includes the digit 8 once, where the digit has a
value of:
a. 8 million
b. 8 thousand
c. 8 hundred
d. 80 thousand
Fill in the missing symbols (< or >).
7,142,294 7,124,294 99,000 600,000
6,090,100 690,100 1,300,610 140,017
589,940 1,010,222
Put these numbers in order from smallest to largest.
8,102,304 8,021,403 843,021 8,043,021
196NPV–3 Numbers up to 10 million in the linear number system
Reason about the location of any number up to 10 million, including decimal fractions, in
the linear number system, and round numbers, as appropriate, including in contexts.
6NPV–3 Teaching guidance
Pupils have already learnt about the location of whole numbers with up to 4 digits in the
linear number system (1NPV–2, 2NPV–2, 3NPV–3 and 4NPV–3) and about the location
of decimal fractions with up to 2 decimal places between whole numbers in the linear
number system (5NPV–3). Pupils must now extend their understanding to larger
numbers.
Pupils need to be able to identify or place numbers with up to 7 digits on marked number
lines with a variety of scales, for example placing 12,500 on a 12,000 to 13,000 number
line, and on a 10,000 to 20,000 number line.
Figure 3: placing 12,500 on a 12,000 to 13,000 number line marked, but not labelled, in multiples of
100
Figure 4: placing 12,500 on a 10,000 to 20,000 number line marked, but not labelled, in multiples of
1,000
Pupils need to be able to estimate the value or position of numbers on unmarked or
partially marked numbers lines, using appropriate proportional reasoning.
Figure 5: estimating the position of 65,000 on an unmarked 50,000 to 100,00 number line
20In the example below, pupils should reason: “a must be about 875,000 because it is
about halfway between the midpoint of the number line, which is 850,000, and 900,000.”
Figure 6: identifying 875,000 on a 800,00 to 900,000 number line marked only with a midpoint
Pupils should understand that, to estimate the position of a number with more significant
digits on a large-value number line, they must attend to the leading digits and can ignore
values in the smaller place-value positions. For example, when estimating the position of
5,192,012 on a 5,100,000 to 5,200,000 number line they only need to attend to the first 4
digits.
Pupils must also be able to round numbers in preparation for key stage 3, when they will
learn to round numbers to a given number of significant figures or decimal places. They
have already learnt to round numbers with up to 4 digits to the nearest multiple of 1,000,
100 and 10, and to round decimal fractions to the nearest whole number or multiple of
0.1. Now pupils should extend this to larger numbers. They must also learn that numbers
are rounded for the purpose of eliminating an unnecessary level of detail. They must
understand that rounding is a method of approximating, and that rounded numbers can
be used to give estimated values including estimated answers to calculations.
Pupils should only be asked to round numbers to a useful and appropriate level: for
example, rounding 7-digit numbers to the nearest 1 million or 100,000, and 6-digit
numbers to the nearest 100,000 or 10,000. Pupils may use a number line for support, but
by the end of year 6, they need to be able to round numbers without a number line. As
with previous year groups (3NPV–3, 4NPV–3 and 5NPV–3), pupils should first learn to
identify the previous and next given multiple of a power of 10, before identifying the
closest of these values. In the examples below, for 5,192,012, pupils must be able to
identify the previous and next multiples of 1 million and 100,000, and round to the
nearest of each.
Figure 7: using a number line to identify the previous and next multiple of 1 million
21Figure 8: using a number line to identify the previous and next multiple of 100,000
Language focus
“The previous multiple of 1 million is 5 million. The next multiple of 1 million is
6 million.”
“The previous multiple of 100,000 is 5,100,000. The next multiple of 100,000 is
5,200,000.”
Figure 9: identifying the nearest multiple of 1 million and the nearest multiple of 100,000
Language focus
“The closest multiple of 1 million is 5 million.”
“5,192,012 rounded to the nearest million is 5 million.”
“The closest multiple of 100,000 is 5,200,000.”
“5,192,012 rounded to the nearest 100,000 is 5,200,000.”
Pupils should explore the different reasons for rounding numbers in a variety of contexts,
such as the use of approximate values in headlines, and using rounded values for
22estimates. They should discuss why a headline, for example, might use a rounded value,
and when precise figures are needed.
Finally, pupils should also be able to count forwards and backwards, and complete
number sequences, in steps of powers of 10 (1, 10, 100, 1,000, 10,000 and 100,000).
Pay particular attention to counting over ‘boundaries’, for example:
• 2,100,000 2,000,000 1,900,000
• 378,500 379,500 380,500
Making connections
Here, pupils must apply their knowledge from 6NPV–1, that each place value unit is
made up of 10 of the unit to its right, to understand how each interval on a number line
or scale is made up of 10 equal parts. This also links to 6NPV–4, in which pupils need
to be able to read scales divided into 2, 4, 5 and 10 equal parts.
6NPV–3 Example assessment questions
Show roughly where each of these numbers is located on the number line below.
2,783,450 7,000,500 5,250,000 8,192,092 99,000
Estimate the values of a, b, c and d.
For each number:
write the previous and next multiple of 1 million
circle the previous or next multiple of 1 million which is closest to the number
23Fill in the missing numbers.
6,361,040 6,371,040 6,381,040 6,391,040 6,401,040 6,411,040
2,004,567 2,003,567 2,002,567 2,001,567 2,000,567 1,999,567
7,730,004 7,930,004 8,030,004 8,230,004 8,430,004
9,149,301 9,129,301 9,119,301 9,089,301
What might the missing number be in this web page?
A swimming pool holds approximately 82,000 litres of water. The capacity of the
swimming pool has been rounded to the nearest multiple of 1,000. Fill in the missing
numbers to complete the sentences.
a. The minimum amount of water that the pool could hold is .
b. The maximum amount of water that the pool could hold is .
246NPV–4 Reading scales with 2, 4, 5 or 10 intervals
Divide powers of 10, from 1 hundredth to 10 million, into 2, 4, 5 and 10 equal parts, and
read scales/number lines with labelled intervals divided into 2, 4, 5 and 10 equal parts.
6NPV–4 Teaching guidance
It is important for pupils to be able to divide powers of 10 into 2, 4, 5 or 10 equal parts
because these are the intervals commonly found on measuring instruments and graph
scales. Pupils have already learnt to divide 1, 100 and 1,000 in this way (5NPV–4,
3NPV–4 and 4NPV–4 respectively), and must now extend this to larger powers of 10.
Pupils should be able to make connections between powers of 10, for example,
describing similarities and differences between the values of the parts when 1 million,
1,000 and 1 are divided into 4 equal parts.
Figure 10: bar models showing 1 million, 1,000 and 1 partitioned into 4 equal parts
Pupils should be able to skip count in these intervals forwards and backwards from any
starting number (for example, counting forward from 800,000 in steps of 20,000, or
counting backwards from 5 in steps of 0.25). This builds on counting in steps of 10, 20,
25 and 50 in year 3 (3NPV–4 ), in steps of 100, 200, 250 and 500 in year 4 (4NPV–4),
and in steps of 0.1, 0.2, 0.25 and 0.5 in year 5 (5NPV–4).
Pupils should practise reading measurement and graphing scales with labelled
power-of-10 intervals divided into 2, 4, 5 and 10 equal parts.
Pupils need to be able to write and solve addition, subtraction, multiplication and division
equations related to powers of 10 divided into 2, 4, 5 and 10 equal parts, as exemplified
for 1 million and 4 equal parts below. Pupils should be able to connect finding equal parts
1
of a power of 10 to finding ,1, 1
or 1
of the value.
2 4 5 10
25750,000 + 250,000 =
1,000,000
1,000,000 – 250,000 = 750,000 1,000,000 – 750,000 = 250,000
1,000,000 ÷ 4 =250,000 1,000,000 ÷ 250,000 =
4
4 × 250,000 =
1,000,000 250,000 × 4 =
1,000,000
1
of 1,000,000 = 250,000
4
Making connections
Dividing powers of 10 into 10 equal parts is also assessed as part of 6NPV–1.
Reading scales also builds on number-line knowledge from 6NPV–3. Conversely,
experience of working with scales with 2, 4, 5 or 10 divisions in this criterion improves
pupils’ estimating skills when working with unmarked number lines and scales as
described in 6NPV–3.
266NPV–4 Example assessment questions
1
If of a 1kg bag of flour is used, how much is left?
10
1
In 2005, the population of Birmingham was about 1 million. At that time, about of
5
the population was over 60 years old. Approximately how many over-60s lived in
Birmingham in 2005?
A builder ordered 1,000kg of sand. She has about 300kg left. What fraction of the
total amount is left?
Fill in the missing parts.
Fill in the missing numbers.
27The bar chart shows the approximate populations of 3 different towns. What are the
populations?
What mass does each scale show?
28Some children are trying to raise £200,000 for charity. The diagram shows how much
they have raised so far.
a. How much money have they raised?
b. How much more money do they need to raise to meet their target?
6AS/MD–1 Quantify additive and multiplicative relationships
Understand that 2 numbers can be related additively or multiplicatively, and quantify
additive and multiplicative relationships (multiplicative relationships restricted to
multiplication by a whole number).
6AS/MD–1 Teaching guidance
Throughout key stage 2, pupils have learnt about and used 2 types of mathematical
relationship between numbers: additive relationships and multiplicative relationships. In
year 6, pupils should learn to represent the relationship between 2 given numbers
additively or multiplicatively, as well as use such a representation to calculate a missing
number, including in measures and statistics contexts.
Consider the following: Holly has cycled 20km. Lola has cycled 60km.
We can describe the relationship between the distances either additively (Lola has cycled
40km further than Holly; Holly has cycled 40km fewer than Lola) or multiplicatively
29(Lola has cycled 3 times the distance that Holly has cycled). The relationship between
the numbers 20 and 60 can be summarised as follows.
Figure 11: additive relationship between 20 and Figure 12: multiplicative relationship between
60 20 and 60
Language focus
“The relationship between 2 numbers can be expressed additively or multiplicatively.”
As pupils progress into key stage 3, the ability to relate, recognise and use multiplicative
relationships is essential. A pupil who can think multiplicatively would, for example,
calculate the cost of 1.2m of ribbon at 75p per metre as 1.2 × 75p , whereas a pupil who
was still thinking only in terms of additive relationships would use the approach of finding
the cost of 0.2m (15p) and adding it to the cost of 1m (75p). During key stage 3, pupils
will regularly use calculators to solve problems with this type of structure, and the
multiplicative approach is more efficient because it involves fewer steps.
Given any 2 numbers (related by a whole-number multiplier), pupils must be able to
identify the additive relationship (in the example above, +40 and −40 ) and the
multiplicative relationship (in the example above ×3 and ÷3 ). Though multiplicative
relationships should be restricted to whole-number multipliers, pupils should be able to
connect division by the whole number to scaling by a unit fraction: in the example above,
this corresponds to understanding that because 60 ÷ 3 = 20 , 20 is one-third times the
size of 60.
When given a sequence of numbers, pupils should be able to identify whether the terms
are all related additively or multiplicatively, identify the specific difference or multiplier and
use this to continue a sequence either forwards or backwards. Pupils will need to use
formal written methods to calculate larger numbers in sequences.
30Figure 13: completing a sequence where the difference between adjacent terms is 7.5
Figure 14: completing a sequence where each term is 5 times the previous
Making connections
In 6AS/MD–4 pupils solve problems with 2 unknowns, where the relationship between
the unknowns may be additive, multiplicative or both, for example: find 2 numbers,
where one is 3 times the size of the other, and the difference between them is 40.
316AS/MD–1 Example assessment questions
Fill in the missing numbers.
300 + 1,200
= 75= 3 + +0.1 =
10
300 × 1,200
= 75= 3 × ×0.1 =
10
Write an expression in each box to show the relationship between numbers 25 and
75. Is there more than one way to answer this question? Explain.
The examples below show the first 2 numbers in a sequence. Find 2 different ways to
continue each sequence, using addition for the first and multiplication for the second.
a. 4 16 or 4 16
b. 2 200 or 2 200
c. 0.01 10 or 0.01 10
Complete these sequences.
0.5 5 9.5 27.5 32
0.5 0.75 1
25 125 625
0.2 6 180
326AS/MD–2 Derive related calculations
Use a given additive or multiplicative calculation to derive or complete a related
calculation, using arithmetic properties, inverse relationships, and place-value
understanding.
6AS/MD–2 Teaching guidance
In previous year groups in key stage 2 pupils have learnt about and used the
commutative and associative properties of addition (3AS–3), and the commutative,
associative and distributive properties of multiplication (4MD–2 and 4MD–3).
Pupils have also implicitly used the compensation property of addition, for example, when
partitioning two-digit numbers in different ways in year 2:
70 + 2 = 72 60 + 12 = 72
In year 6, pupils should learn the compensation property of addition.
Language focus
“If one addend is increased and the other is decreased by the same amount, the sum
stays the same.”
Pupils should be able to use the compensation property of addition to complete
equations such as 25 + 35= 27.5 + ? , and to help them solve calculations such as
27.5 + 32.5.
Similarly, pupils may have implicitly used the compensation property of multiplication, for
example, when recognising connections between multiplication table facts:
5 × 8 = 10 × 4
In year 6, pupils should learn the compensation property of multiplication.
Language focus
“If I multiply one factor by a number, I must divide the other factor by the same number
for the product to stay the same.”
33Pupils should be able to use the compensation property of multiplication to complete
equations such as 0.3 × 320 = 3 × ? , and to help them solve calculations such as
0.3 × 320 .
Pupils have extensive experience about the effect on the product of scaling one factor
from 3NF–3, 4F–3 and 5NF–2, where they learnt to scale known number facts by 10,
100, one-tenth and one-hundredth. Now they can generalise.
Language focus
“If I multiply one factor by a number, and keep the other factor the same, I must
multiply the product by the same number.”
Pupils should practise combining their knowledge of arithmetic properties and
relationships to solve problems such as the examples here and in the Example
assessment questions below.
Example problem 1 Example problem 2
Question: Explain how you would use Question: Explain how you would use
the first equation to complete the the first equation to complete the
second equation: second equation:
2,448 ÷ 34 =
72 921
= 349 + 572
72 × 24,480
= 92.1
= 44.9 +
Explanation: Explanation:
Use the inverse relationship Apply understanding of place value,
between multiplication and division making the sum and addends 1
to restate the equation: tenth times the size.
72 × 34 =
2,448 92.1
= 34.9 + 57.2
Apply the compensation property of
Apply understanding of place-value:
addition to solve the equation: add
the product can be made 10 times
10 to the first addend and subtract
the size by making one of the
10 from the second addend.
factors 10 times the size.
92.1
= 44.9 + 47.2
72 × 340 =
24,480
Pupils should learn to write a series of written equations to justify their solutions.
Being able to work fluently with related equations in this way will prepare pupils for
manipulating algebraic equations in key stage 3 and writing proofs.
34Pupils can already apply place-value understanding to known multiplication facts to scale
one factor, for example, 3 × 4 =
12 , so 3 × 40 =
120 . Now they should extend this to
12 , so 30 × 40 =
scaling both factors, for example, 3 × 4 = 1,200 .
Making connections
In this criterion, pupils use their understanding from 6NPV–1 of scaling numbers by 10,
100 and 1,000.
6AS/MD–2 Example assessment questions
Fill in the missing numbers.
327 + 278 = 330 + 25 × 48 = 50 ×
327 + 515 = 842
Use this calculation to complete the following equations.
+61.5 =
84.2
8,420 – = 3,270
85,200 – 52,500 =
21,760
= 256 × 85
Use this calculation to complete the following equations.
256 × 8.5 =
2,560 × 85 =
2,156 ÷ 85 =
3,128 ÷ 23 =
136
Use the division calculation so solve the following calculation. Explain your answer.
24 × 136 =
Fill in the missing number.
25 × 60
= ×60 + 120
356AS/MD–3 Solve problems involving ratio relationships
Solve problems involving ratio relationships.
6ASMD–3 Teaching guidance
Pupils already have the arithmetic skills to solve problems involving ratio. They should
now learn to describe 1-to-many (and many-to-1) correspondence structures.
Language focus
“For every 1 cup of rice you cook, you need 2 cups of water.”
“For every 10 children on the school trip, there must be 1 adult.”
Pupils should learn to complete ratio tables, given a 1-to-many or many-to-1 relationship.
cups of
1 2 3 4 5 6
rice
cups of
2 4 6 8 10 12
water
number of
10 20 30 40 50 60
children
number of
1 2 3 4 5 6
adults
Pupils must recognise that proportionality is preserved in these contexts, for example,
there is always twice the volume of water needed compared to the volume or rice,
regardless of how much rice there is. This will prepare pupils for key stage 3, when they
will learn to describe correspondence structures using ratio notation and to express ratios
in their simplest forms.
Pupils should be able to recognise a 1-to-many or many-to-1 structure, without it being
explicitly given and use the relationship to solve problems. For example, here pupils
should recognise that, in both examples, for every 1 red bead there are 3 blue beads (or
for every 3 blue beads there is 1 red bead), irrespective of the arrangement of the beads.
36Figure 15: bead strings, each with the structure ‘for every 1 red bead, there are 3 blue beads’
For examples like this, pupils should also be able to include the total quantity in a table.
number of red beads 1 2 3 4
number of blue beads 3 6 9 12
total number of beads 4 8 12 16
Pupils should also be able to answer questions such as:
• if there were 5 red beads, how many blue beads would there be?
• if there were 21 blue beads, how many beads would there be altogether?
• if there were 40 beads altogether, how many red beads and how many blue beads
would there be?
Pupils must also learn to describe and solve problems related to many-to-many
structures.
Language focus
“For every 2 yellow beads there are 3 green beads”.
Pupils may initially use manipulatives, such as cubes or beads, for support, but by the
end of year 6, they must be able to complete many-to-many correspondence tables and
solve related problems without manipulatives.
number of yellow beads 2 4 6 8
number of green beads 3 6 9 12
total number of beads 5 10 15 20
Pupils should also begin to prepare for using the unitary method at key stage 3, when it is
required for unit conversions, percentage calculations and other multiplicative problems.
For example, if they are given a smoothie recipe for 2 people (20 strawberries, 1 banana
and 150ml milk), they should be able to adjust the recipe by multiplying or dividing by a
37whole number, for example, dividing the quantities by 2 to find the amounts for 1 person,
or multiplying the quantities by 3 to find the amounts for 6 people. At key stage 3, pupils
would then, for example, be able to use the unitary method to adjust the recipe for 5
people, via calculating the amounts for 1 person.
Making connections
To recognise a one-to-many or many-to-one structure, pupils need to be able to
identify multiplicative relationships between given numbers (6AS/MD–2).
6AS/MD–3 Example assessment questions
For every 1 litre of petrol, Miss Smith’s car can travel about 7km.
a. How many kilometres can Miss Smith’s car travel on 6 litres of petrol?
b. Miss Smith lives about 28km from school. How many litres of petrol does she use
to get to school?
For every 3m of fence I need 4 fence panels. The fence will be 15m long. How many
fence panels will I need?
I am decorating a cake with fruit. I use 2 raspberries for every 3 strawberries.
Altogether I put 30 berries on the cake.
a. How many raspberries did I use?
b. How many strawberries did I use?
For every 500g of excess baggage I take on an aeroplane, I must pay £7.50. I have
3.5kg of excess baggage. How much must I pay?
38Lily and Ralph are eating grapes. The diagram represents the relationship between
the number of grapes that the children eat.
Fill in the missing numbers.
Number of Number of
grapes that Lily grapes that
eats Ralph eats
1
20
3
Giya is planting flowers in her garden. For every 5 red flowers she plants, she plants 3
yellow flowers. If Giya plants 18 yellow flowers, how many red flowers does she
plant?
I am making a necklace. So far, it has 4 black beads and 1 white bead. How many
more white beads would I need to add so that there are 4 white beads for every 1
black bead?
6AS/MD–4 Solve problems with 2 unknowns
Solve problems with 2 unknowns.
6AS/MD–4 Teaching guidance
Pupils need to be able to solve problems with 2 unknowns where:
• there are an infinite number of solutions
• there is more than 1 solution
• there is only 1 solution
Pupils may have seen equations with 2 unknowns before, for example, when recognising
connections between multiplication table facts:
5× =10 ×
39In year 6, pupils must recognise that an equation like this has many (an infinite number)
of solutions. They should learn to provide example solutions by choosing a value for one
unknown and then calculating the other unknown.
Pupils should be able to solve similar problems where there is more than one solution,
but not an infinite number, for example:
Danny has some 50p coins and some 20p coins. He has £1.70 altogether. How many of
each type of coin might he have?
In these cases, pupils may choose a value for the first unknown and be unable to solve
the equation for the other unknown (pupils may first set the number of 50p pieces at 2,
giving £1, only to find that it is impossible to make up the remaining 70p from 20p coins).
Pupils should then try a different value until they find a solution. For a bound problem
with only a few solutions, like the coin example, pupils should be able to find all possible
solutions by working systematically using a table like that shown below. They should be
able to reason about the maximum value in each column.
Figure 16: finding the 2 solutions to the coin problem: one 50p coin and six 20p coins, or three 50p
coins and one 20p coin
Pupils must also learn to solve problems with 2 unknowns that have only 1 solution.
Common problems of this type involve 2 pieces of information being given about the
relationship between the 2 unknowns – 1 piece of additive information and either another
piece of additive information or a piece of multiplicative information. Pupils should learn
to draw models to help them solve this type of problem.
40Example problem 1 Example problem 2
Question: The sum of 2 numbers is 25, Question: The sum of 2 numbers is 48.
and the difference between them is 7. One number is one-fifth times the size
What are the 2 numbers? of the other number. What are the 2
numbers?
Solution: Solution:
Figure 17: using a bar model to solve a Figure 18: using a bar model to solve a
problem with 2 unknowns – example 1 problem with 2 unknowns – example 2
a = 9 + 7 = 16 a=8
b=9 b = 5×8 =40
The numbers are 16 and 9. The numbers are 8 and 40.
Pupils should also be able to use bar modelling to solve more complex problems with 2
unknowns and 1 solution, such as: 4 pears and 5 lemons cost £3.35. 4 pears and 2
lemons cost £2.30. What is the cost of 1 lemon?
Figure 19: using a bar model to solve a problem with 2 unknowns – example 3
41Solving problems with 2 unknowns and 1 solution will prepare pupils for solving
simultaneous equations in key stage 3.
Pupils should practise solving a range of problems with 2 unknowns, including contextual
measures and geometry problems.
Making connections
Within this criterion, pupils must be able to use their understanding of how 2 numbers
can be related additively or multiplicatively (6AS/MD–1). In 6G–1 pupils solve geometry
problems with 2 unknowns, for example, finding the unknown length and unknown
width of a rectangle with a perimeter of 14cm.
6AS/MD–4 Example assessment questions
A baker is packing 60 cakes into boxes. A small box can hold 8 cakes and a large box
can hold 12 cakes.
a. How many different ways can he pack the cakes?
b. How can he pack the cakes with the fewest number of boxes?
1 eraser and 5 pencils cost a total of £3.35.
5 erasers and 5 pencils cost a total of £4.75.
a. How much does 1 eraser cost?
b. How much does 1 pencil cost?
An adult ticket for the zoo costs £2 more than a child ticket. I spend a total of £33
buying 3 adult and 2 child tickets.
a. How much does an adult ticket cost?
b. How much does a child ticket cost?
The balances show the combined masses of some large bags of dog food and some
small bags of dog food.
How much does each bag-size cost?
A rectangle with side-lengths a and b has a perimeter of 30cm. a is a 2-digit whole
number and b is a 1-digit whole number. What are the possible values of a and b?
42The diagram shows the total cost of the items in each row and column. Fill in the 2
missing costs.
6F–1 Simplify fractions
Recognise when fractions can be simplified, and use common factors to simplify
fractions.
6F–1 Teaching guidance
In year 5, pupils learnt to find equivalent fractions (5F–2). Now pupils must build on this
and learn to recognise when fractions are not in their simplest form. They should use
their understanding of common factors (5MD–2) to simplify fractions.
Pupils should learn that when the numerator and denominator of a fraction have no
common factors (other than 1) then the fraction is in its simplest form. Pupils should learn
that a fraction can be simplified by dividing both the numerator and denominator by a
common factor. They must realise that simplifying a fraction does not change its value,
and the simplified fraction has the same position in the linear number system as the
original fraction. Pupils should begin with fractions where the numerator and denominator
6
have only one common factor other than 1, for example .
15
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